Apparently Mod Podge is a brand of craft products and adhesives: https://mod-podge.com/
Apparently Mod Podge is a brand of craft products and adhesives: https://mod-podge.com/
Having a soft base helps when transporting. I once had to move a 9.000 piece puzzle - and without some bending it would have been difficult getting it into the car.
For the biggest jigsaws out there it gets difficult to find a good wall for it. Weight and bending of wood or MDF can become an issue. And even some walls are not as flat as they look like...
Further considerations:
Multi-ply is frequently available in 5'x5' sheets and can be tough to find in 4'x8's. MDF is universally 49"x97" (so you can net a 4'x8' after accounting for the dinged up edges). The size of your puzzle may inform your choice.
Home Depot et al. will usually cut a sheet to size on their panel saw. Your local lumberyard may or may not be interested in doing so for a small order.
Home Depot et al. generally aren't interested in delivering for an even half reasonable price if they do it at all. A real lumberyard might be willing to do so, though it's unlikely to be cheap for a single sheet order ($25 for an order under $500, last I needed a delivery. Cheaper than owning a truck or van for me). They also usually dispatch trucks on a schedule. Mine goes north on Tuesdays and south on Thursdays, for example. Plan ahead and don't expect your stuff to show up like Amazon packages the next day.
For the 9.000s I always used paper for the backside (old calendars are a good source for big sheets with excellent quality). Using one wooden lath at the top was enough to ensure stability for a 18.000. The 9.000s didn't need that - just paper and a bit of duct-tape on the backside were enough.
I found that a coating of quick-drying, low-viscosity liquid glue on the front side works pretty well, because the glue easily flows into the gaps between the pieces, and then into the space between the puzzle and the board. Once the glue dries off, all the pieces and the backing board become a single chunk of cellulose-infused plastic. This is probably much better than either the thick mod podge or the adhesive sheet that OP tried to use at first.
Or sheet of hardboard and a stanley knife, which is probably the most sensible option in context.
All in all when I needed a one-off cut I used a box cutter knife, and when I needed a series I used a circular saw with a plywood blade. The setup is similar (needs an offset guide) but the cut is much straighter, faster, and less error prone. I think the only time I'd use a jigsaw is for making non-straight cuts like some curved artistic shape or something.
(Like mentioned I did have it in the reply as well)
Proper track saws though tend to come with finer tooth blades with a thinner kerf so that they cut cleaner than a circular saw. They also retract within a kind of housing so that they're safer — only plunging down when you lean on them to make the cut.
Once you use one, you'll not go back to your circular saw unless, for example, you're trying to cut some framing lumber in place.
I have a Makita [1] which I like a lot, but my sister, wanting to save a little dosh, got a Kreg [2] and is happy with it.
[1] https://www.amazon.com/Makita-SP6000J1-Plunge-Circular-Stack...
[2] https://www.amazon.com/Kreg-ACS-SAWBB-Plunge-Cut-replaces-AC...